Autoregressive Research Papers - Academia.edu (original) (raw)

A well-known and widely used model for estimation in econometrics is the so called Generalized Least Squares model, shortly the GLS model. It is specified as follows: , where is a Tx1 vector of observations, a Txk matrix of explanatory variables, a kx1 vector of parameters to be estimated and a Tx1 vector of disturbances. The disturbance vector is distributed with mean zero or = 0 and '=V, a positive definite matrix. The symbol ' is used to denote the transpose of a vector or matrix. In case is zero, the model reduces to , the common or pure time series model. For our purpose we specify the disturbance term as =-+ + , t= 1,2,…T (1) where is white noise: = 0 and '= I. Here the first part, the part, is called the autoregressive or AR part, the second part, with parameter , is the moving average or MA part. When both parameters are specified, we speak of an ARMA model. If the vector is estimated by the so-called Aitken estimator for every choice of V. By defining , we can reformulate the problem as finding those values for and for which is minimized. So far the minimum distance approach. If one prefers a maximum likelihood approach one has to minimize , where | | is the determinant of the covariance matrix V. Of course the first part, , converges rapidly to 1, when T becomes large – under the commonly used assumptions. This makes clear that the main problem is a handsome expression for the covariance matrix and its inverse. 2. The ARMA covariance matrix in closed form Now we will present an expression of the ARMA covariance matrix in closed form. It is easy to invert and to differentiate. We will avoid most of the technical details, these can be find in the literature. First we will rewrite equation (1) in matrix form (Pagan, 1974). Next we will derive a matrix equation from which has to be derived. As appears at both sides of this equation, the solution is not straightforward, although the solution is – maybe – surprisingly simple. The start is the definition of a (special) Toeplitz matrix: a lower band matrix of order x with elements